Question: $\dfrac{ -8m - 6n }{ -7 } = \dfrac{ -6m + 9p }{ 6 }$ Solve for $m$.
Solution: Multiply both sides by the left denominator. $\dfrac{ -8m - 6n }{ -{7} } = \dfrac{ -6m + 9p }{ 6 }$ $-{7} \cdot \dfrac{ -8m - 6n }{ -{7} } = -{7} \cdot \dfrac{ -6m + 9p }{ 6 }$ $-8m - 6n = -{7} \cdot \dfrac { -6m + 9p }{ 6 }$ Multiply both sides by the right denominator. $-8m - 6n = -7 \cdot \dfrac{ -6m + 9p }{ {6} }$ ${6} \cdot \left( -8m - 6n \right) = {6} \cdot -7 \cdot \dfrac{ -6m + 9p }{ {6} }$ ${6} \cdot \left( -8m - 6n \right) = -7 \cdot \left( -6m + 9p \right)$ Distribute both sides ${6} \cdot \left( -8m - 6n \right) = -{7} \cdot \left( -6m + 9p \right)$ $-{48}m - {36}n = {42}m - {63}p$ Combine $m$ terms on the left. $-{48m} - 36n = {42m} - 63p$ $-{90m} - 36n = -63p$ Move the $n$ term to the right. $-90m - {36n} = -63p$ $-90m = -63p + {36n}$ Isolate $m$ by dividing both sides by its coefficient. $-{90}m = -63p + 36n$ $m = \dfrac{ -63p + 36n }{ -{90} }$ All of these terms are divisible by $9$ Divide by the common factor and swap signs so the denominator isn't negative. $m = \dfrac{ {7}p - {4}n }{ {10} }$